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Theorem bcthlem2 25282

Description: Lemma for bcth 25286. The balls in the sequence form an inclusion chain. (Contributed by Mario Carneiro, 7-Jan-2014.)

**Hypotheses**
Ref	Expression
bcth.2	⊢ 𝐽 = (MetOpen‘𝐷)
bcthlem.4	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
bcthlem.5	⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
bcthlem.6	⊢ (𝜑 → 𝑀:ℕ⟶(Clsd‘𝐽))
bcthlem.7	⊢ (𝜑 → 𝑅 ∈ ℝ⁺)
bcthlem.8	⊢ (𝜑 → 𝐶 ∈ 𝑋)
bcthlem.9	⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
bcthlem.10	⊢ (𝜑 → (𝑔‘1) = ⟨𝐶, 𝑅⟩)
bcthlem.11	⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))

**Assertion**
Ref	Expression
bcthlem2	⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Distinct variable groups: 𝑘,𝑛,𝑟,𝑥,𝑧 𝐶,𝑟,𝑥 𝑔,𝑘,𝑛,𝑟,𝑥,𝑧,𝐷 𝑔,𝐹,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝐽,𝑘,𝑛,𝑟,𝑥,𝑧 𝑔,𝑀,𝑘,𝑛,𝑟,𝑥,𝑧 𝜑,𝑘,𝑛,𝑟,𝑥,𝑧 𝑥,𝑅 𝑔,𝑋,𝑘,𝑛,𝑟,𝑥,𝑧 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑧,𝑔,𝑘,𝑛) 𝑅(𝑧,𝑔,𝑘,𝑛,𝑟)

**Proof of Theorem bcthlem2**
Step	Hyp	Ref	Expression
1		bcthlem.11	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)))
2		fvoveq1 7433	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑔‘(𝑘 + 1)) = (𝑔‘(𝑛 + 1)))
3		id 22	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → 𝑘 = 𝑛)
4		fveq2 6881	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑔‘𝑘) = (𝑔‘𝑛))
5	3, 4	oveq12d 7428	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘𝐹(𝑔‘𝑘)) = (𝑛𝐹(𝑔‘𝑛)))
6	2, 5	eleq12d 2829	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ↔ (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛))))
7	6	rspccva 3605	. . . . 5 ⊢ ((∀𝑘 ∈ ℕ (𝑔‘(𝑘 + 1)) ∈ (𝑘𝐹(𝑔‘𝑘)) ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
8	1, 7	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)))
9		bcthlem.9	. . . . . 6 ⊢ (𝜑 → 𝑔:ℕ⟶(𝑋 × ℝ⁺))
10	9	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))
11		bcth.2	. . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷)
12		bcthlem.4	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))
13		bcthlem.5	. . . . . . 7 ⊢ 𝐹 = (𝑘 ∈ ℕ, 𝑧 ∈ (𝑋 × ℝ⁺) ↦ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑟 < (1 / 𝑘) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑟)) ⊆ (((ball‘𝐷)‘𝑧) ∖ (𝑀‘𝑘))))})
14	11, 12, 13	bcthlem1 25281	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ (𝑔‘𝑛) ∈ (𝑋 × ℝ⁺))) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
15	14	expr 456	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘𝑛) ∈ (𝑋 × ℝ⁺) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))))
16	10, 15	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑛𝐹(𝑔‘𝑛)) ↔ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)))))
17	8, 16	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))))
18		cmetmet 25243	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
19	12, 18	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
20		metxmet 24278	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
22	11	mopntop 24384	. . . . . . . . . 10 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ Top)
24		xp1st 8025	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋)
25		xp2nd 8026	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ⁺)
26	25	rpxrd 13057	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)
27	24, 26	jca 511	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*))
28		blssm 24362	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
29	28	3expb 1120	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((1^st ‘(𝑔‘(𝑛 + 1))) ∈ 𝑋 ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) ∈ ℝ^*)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
30	21, 27, 29	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) ⊆ 𝑋)
31		df-ov 7413	. . . . . . . . . . . 12 ⊢ ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
32		1st2nd2 8032	. . . . . . . . . . . . 13 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → (𝑔‘(𝑛 + 1)) = ⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩)
33	32	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) = ((ball‘𝐷)‘⟨(1^st ‘(𝑔‘(𝑛 + 1))), (2^nd ‘(𝑔‘(𝑛 + 1)))⟩))
34	31, 33	eqtr4id 2790	. . . . . . . . . . 11 ⊢ ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
35	34	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((1^st ‘(𝑔‘(𝑛 + 1)))(ball‘𝐷)(2^nd ‘(𝑔‘(𝑛 + 1)))) = ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))))
36	11	mopnuni 24385	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
37	21, 36	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
38	37	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → 𝑋 = ∪ 𝐽)
39	30, 35, 38	3sstr3d 4018	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽)
40		eqid 2736	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
41	40	sscls 22999	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ∪ 𝐽) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
42	23, 39, 41	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))))
43		difss2 4118	. . . . . . . 8 ⊢ (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
44		sstr2 3970	. . . . . . . 8 ⊢ (((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
45	42, 43, 44	syl2im 40	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
46	45	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺)) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))))
47	46	ex 412	. . . . 5 ⊢ (𝜑 → ((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) → ((2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) → (((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛)) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))))
48	47	3impd 1349	. . . 4 ⊢ (𝜑 → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
49	48	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝑔‘(𝑛 + 1)) ∈ (𝑋 × ℝ⁺) ∧ (2^nd ‘(𝑔‘(𝑛 + 1))) < (1 / 𝑛) ∧ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑔‘(𝑛 + 1)))) ⊆ (((ball‘𝐷)‘(𝑔‘𝑛)) ∖ (𝑀‘𝑛))) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛))))
50	17, 49	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))
51	50	ralrimiva 3133	1 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝑔‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝑔‘𝑛)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3052 ∖ cdif 3928 ⊆ wss 3931 ⟨cop 4612 ∪ cuni 4888 class class class wbr 5124 {copab 5186 × cxp 5657 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 1^st c1st 7991 2^nd c2nd 7992 1c1 11135 + caddc 11137 ℝ^*cxr 11273 < clt 11274 / cdiv 11899 ℕcn 12245 ℝ⁺crp 13013 ∞Metcxmet 21305 Metcmet 21306 ballcbl 21307 MetOpencmopn 21310 Topctop 22836 Clsdccld 22959 clsccl 22961 CMetccmet 25211

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-topgen 17462 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-top 22837 df-topon 22854 df-bases 22889 df-cld 22962 df-cls 22964 df-cmet 25214

This theorem is referenced by: bcthlem3 25283 bcthlem4 25284

W3C validator